Week1 Bayesian Network

Introduction¶

Graphical Models¶

• Variables are represented by nodes

Graphcial representation¶

• Intuitive & Compact data structure
• Efficient reasoning using general-purpose algorithms
• Sparse Parameterization
  • feasible elicitation
  • learning from data

Applicationd¶

Overview¶

Representation¶

- Directed and Undirected
- Temporal and Plate Models

Inference¶

- Exact and Approximate
- Decision Making

Learning¶

- Parameters and Structure
- With and Without Complete data

Probability Distribution¶

Conditional Distribution¶

- Quiz

Marginal Distribution¶

Factor¶

• Factor is a function of variables

• Scope is a set of variables

• Joint Distribution is a factor
  $P(I,D,G) = \phi(I,D,G)$

• Conditional distribution is also a factor

• Facotr is not always about probabilities
  

Factor Product¶

$\phi_1(A,B) * \phi_2(B,C) = \phi_3(A,B,C)$

• Quiz

Factor Marginalization¶

$\sum_B \phi_1(A,B,C) = \phi_2(A,C)$

Factor Reduction¶

• $\phi_1(C=1|A,B)$
• Like the conditional probability

Bayesian Network¶

• A Bayesian network is:
  • A directed acyclic graph(DAG)G whose nodes represent the random variables $X_1,...,X_n$
  • For each node $X_i$, we have a Conditional Probability Distribution(CPD) $P(X_i|Par_G(X_i))$
• The BN represents a joint distribution via the chain rule for bayesian networks $P(X1,...,X_n) = \prod_i P(X_i|Par_G(X_i))$

Chain Rule for BN¶

• Example：
  

• Quiz
  

Prove BN¶

1. we wanto prove Bayesian network is a valid probability (joint) distribution

• $> 0$ ,
• $\sum=1$

Flow of Probabilistic Influence¶

• Without evidence
• With evidence

Active Trails¶

• X→W←Y (V-Structure) is the only way to block a trail

• Quiz

• 1
• 2
• 3
  

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